New tools for the systematic analysis and visualization of electronic excitations. II.ApplicationsFelix Plasser, Stefanie A. Bäppler, Michael Wormit, and Andreas Dreuw Citation: The Journal of Chemical Physics 141, 024107 (2014); doi: 10.1063/1.4885820 View online: http://dx.doi.org/10.1063/1.4885820 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in New tools for the systematic analysis and visualization of electronic excitations. I. Formalism J. Chem. Phys. 141, 024106 (2014); 10.1063/1.4885819 Tracing molecular electronic excitation dynamics in real time and space J. Chem. Phys. 132, 144302 (2010); 10.1063/1.3353161 Experimental investigation of the Jahn-Teller effect in the ground and excited electronic states of the tropylradical. Part II. Vibrational analysis of the A E 3 ″ 2 - X E 2 ″ 2 electronic transition J. Chem. Phys. 128, 084311 (2008); 10.1063/1.2829471 Electronic spectroscopy of toluene–rare-gas clusters: The external heavy atom effect and vibrationalpredissociation J. Chem. Phys. 122, 194315 (2005); 10.1063/1.1899155 The excited electronic states of H 2 CSi J. Chem. Phys. 107, 8823 (1997); 10.1063/1.475174

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THE JOURNAL OF CHEMICAL PHYSICS 141, 024107 (2014)

New tools for the systematic analysis and visualization of electronicexcitations. II. Applications

Felix Plasser,a) Stefanie A. Bäppler, Michael Wormit, and Andreas DreuwInterdisciplinary Center for Scientific Computing, Ruprecht-Karls-University, Im Neuenheimer Feld 368,69120 Heidelberg, Germany

(Received 10 May 2014; accepted 18 June 2014; published online 10 July 2014)

The excited states of a diverse set of molecules are examined using a collection of newly implementedanalysis methods. These examples expose the particular power of three of these tools: (i) naturaldifference orbitals (the eigenvectors of the difference density matrix) for the description of orbitalrelaxation effects, (ii) analysis of the one-electron transition density matrix in terms of an electron-hole picture to identify charge resonance and excitonic correlation effects, and (iii) state-averagednatural transition orbitals for a compact simultaneous representation of several states. Furthermore,the utility of a wide array of additional analysis methods is highlighted. Five molecules with diverseexcited state characteristics are chosen for these tasks: pyridine as a prototypical small heteroaro-matic molecule, a model system of six neon atoms to study charge resonance effects, hexatriene inits neutral and radical cation forms to exemplify the cases of double excitations and spin-polarization,respectively, and a model iridium complex as a representative metal organic compound. Using theseexamples a number of phenomena, which are at first sight unexpected, are highlighted and theirphysical significance is discussed. Moreover, the generality of the conclusions of this paper is veri-fied by a comparison of single- and multireference ab initio methods. © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4885820]

I. INTRODUCTION

In the first part of this series1 (henceforth denoted Pa-per I1) a detailed discussion of a number of excited stateanalysis methods based on density matrices was presentedand their implementation within the algebraic diagrammaticconstruction (ADC) of the polarization propagator2 as imple-mented within the Q-Chem program package3–5 was reported.In this paper, selected applications will be presented, espe-cially pointing out the power of three methods:! Natural difference orbitals (NDO, the eigenvectors of

the difference density matrix) for the description of or-bital relaxation effects.! Analysis of the one-electron transition density matrix(1TDM) in terms of the electron-hole pair represen-tation of the excitation to elucidate charge resonanceeffects.! State-averaged natural transition orbitals (SA-NTOs)to visualize a number of excited states simultaneously.

Examples will also be presented for natural orbitals(NOs)6 and natural transition orbitals (NTOs)7–9 as well asdensities and transition densities in spin-restricted and un-restricted cases. Different numerical descriptors as definedin Paper I1 will be examined including the squared norm !

of the 1TDM, the integral over the attachment/detachmentdensities p, the number of non-vanishing NTO eigenvalues

a)Electronic mail: felix.plasser@iwr.uni-heidelberg.de. URL:http://www.iwr.uni-heidelberg.de/groups/compchem/

PRNT O , and the nu and nu,nl measures of effectively unpairedelectrons.

Furthermore, a new implementation of some of thesemethods for multireference configuration interaction (MR-CI) wavefunctions10–12 within the Columbus programpackage13, 14 is reported. This allows for comparison of resultsstemming from entirely different types of quantum chemicalmethods.

Five distinct example molecules (Fig. 1) will be usedto illustrate different aspects of the methods. First, pyridinewill be considered as a prototypical heteroaromatic moleculewith excited states similar to a large number of interestingmolecules including DNA bases. In this case relaxation ef-fects of nπ∗ states, and static correlation of ππ∗ states will behighlighted. Second, an artificial model system of six neonatoms arranged in a regular hexagon will be employed tostudy the influence of varying interchromophore interactionstrength on electronic coherence in a systematic way. It willbe shown that our methods readily allow for a quantificationof charge resonance effects irrespective of complete orbitaldelocalization. Furthermore, to illuminate the performance ofthe analysis tools in the case of doubly excited states, the ex-citations of hexatriene will be investigated. Methods for ana-lyzing spin-unrestricted wavefunctions will be presented uti-lizing the example of the hexatriene radical cation. Finally, acollection of different excited state analysis methods will beapplied to the excited states of the model iridium complex fac-tris(3-iminoprop-1-en-1-ido)iridum abbreviated Ir(C3H4N)3.In this case, it will be demonstrated that in particular for chal-lenging cases a detailed methodological analysis is indeed ex-tremely beneficial.

0021-9606/2014/141(2)/024107/12/$30.00 © 2014 AIP Publishing LLC141, 024107-1

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024107-2 Plasser et al. J. Chem. Phys. 141, 024107 (2014)

FIG. 1. Molecular systems studied in this work: (a) pyridine, (b) a hexagonof six neon atoms arranged at variable distance d, (c) hexatriene in itsneutral closed-shell and radical cation forms, and (d) the iridium complexIr(C3H4N)3.

II. COMPUTATIONAL DETAILS

The excited states of pyridine were computed at theADC(2)15 level using the cc-pVDZ16 basis set. These compu-tations were performed using C2v symmetry, and the moleculewas positioned in the xz-plane and the N-atom on the z-axis.For comparison, MR-CI10–12 computations were performedusing the cc-pVDZ basis set as well. For this purpose a ref-erence space of 6 electrons in 5 active orbitals (2π , n, 2π∗)was chosen and all single and double excitations out of thesereferences were included in the wavefunction expansion (MR-CISD(6/5)). The orbitals were constructed through a completeactive space self-consistent field (CASSCF) calculation us-ing the same space. MR-CISD energies are reported with theextensivity correction suggested by Pople et al. (+P).17 Thedensity matrices derive from the original MR-CISD wave-functions, i.e., they properly account for static and dynamiccorrelation effects but are not corrected for extensivity. Itmay be noted here that a fully consistent treatment, includ-ing extensivity corrections of the density matrices, is avail-able through the multireference averaged quadratic coupledcluster method, which was chosen in Refs. 18 and 19. But thesimpler approach used here should certainly be sufficient withrespect to all the qualitative and semi-quantitative argumentsgiven later.

The Ne6 system was treated at the ADC(2) level using theaug-cc-pVDZ basis set.16 The six neon atoms were arrangedaccording to a regular hexagon in the xy-plane in a D6h sym-metric configuration. The calculations were performed usingthe commutative D2h subgroup.

For the neutral hexatriene system the excited states werecomputed at the ADC(3) level20, 21 to properly include doubleexcitations.22 The density matrices were then obtained fromcontracting the ADC(3) vectors with the second order inter-mediate state representation.23 In the case of the radical cationthe UADC(2) method was used following Ref. 24. In bothcases the 6-31G*25 basis set was used. C2h symmetry was em-ployed. MR-CISD computations considered a reference spaceof 6 electrons in 6 orbitals (3π , 3π∗) constructed by CASSCFusing the same active space.

Computations on Ir(C3H4N)3 were performed at theADC(2) level. The iridium atom was described with theLANL2DZ effective core potential (in its “small-core” ver-sion) and the corresponding basis set for the active (5s, 5p,5d, 6s, 6p) orbital shells,26 while for the remaining atoms the6-31G* basis set was employed. The frozen core approxima-tion was used for the 1s orbitals on second row atoms whilethe (5s, 5p) “semi-core” orbitals on iridium were kept activein the ADC calculations.

Geometries were generally optimized at the MP2 level,except for Ir(C3H4N)3 where a DFT/PBE0 geometry wasused. All ADC calculations were performed with a develop-ment version of the Q-Chem 4.1 package.3–5 MR-CISD calcu-lations were done within Columbus.13, 14 For post-processingof the 1TDM analysis the “Wave Function Analysis Tools”package27 was employed. The different tools described hereare in part already available for public download and the re-lease of the remaining components is currently in preparation.

III. nπ∗ AND ππ∗ STATES IN PYRIDINE

Pyridine is chosen as a representative of heteroaromaticmolecules, possessing low-lying nπ∗ and ππ∗ states. While ageneral discussion of the excited states of this molecule has al-ready been provided in the literature,28 this work is focused ona detailed analysis of relaxation and correlation effects. De-spite the presumably simple electronic states of this molecule,several features can be highlighted, which are usually not con-sidered when discussing these types of molecules. In particu-lar, ! the nπ∗ states feature strong relaxation effects as mea-

sured by the promotion number p (p ≫ 1, correspond-ing to the integral over the attachment or detachmentdensities) and! the ππ∗ states exhibit static electron-hole correlation(indicated by PRNT O ≫ 1).

To represent the electronic excitations, the state-averagednatural transition orbitals (SA-NTOs), as introduced inPaper I,1 computed for the first six singlet excited states arepresented in Figure 2. They are labeled by the symbol of theirreducible representation and a subscript H or E indicatingwhether these are hole or particle (electron) orbitals. SA-NTOs are constructed as eigenvectors of the state-averagedhole and particle densities, and are intended to provide a com-pact representation of several excited states, which is inde-pendent of the Hartree-Fock (HF) orbital resolution. In thepresent case, due to the high symmetry and the small basis set,the SA-NTOs are almost indistinguishable from the HF fron-tier orbitals (see Figure S1 of the supplementary material29

and Ref. 28). By employing a larger basis set, however, the sit-uation changes completely as is illustrated in Figure S2 of thesupplementary material29 for the HF/aug-cc-pVDZ case. Inthis case the occupied orbitals do still resemble the cc-pVDZones, but on the virtual side there are five diffuse orbitals en-ergetically below the first π∗ one rendering the analysis ofthe excited states in terms of canonical HF orbitals quite te-dious. By contrast, the SA-NTOs at the aug-cc-pVDZ level

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024107-3 Plasser et al. J. Chem. Phys. 141, 024107 (2014)

Hole Particle

ψa1,H (1.79) ψa2,E (2.53)

ψa2,H (1.68) ψb2,E (2.49)

ψb2,H (1.64)

FIG. 2. State-averaged natural transition orbitals for the first six excitedstates of pyridine at the ADC(2)/cc-pVDZ level of theory. Summed ampli-tudes (λi,H , λi,E ) are given in parentheses.

(Figure S3 of the supplementary material29) closely resemblethe cc-pVDZ ones (Figure 2).

The five frontier orbitals naturally give rise to six differ-ent excited states, whose properties are listed in Table I. Thefirst two excited states possess nπ∗ character while the re-maining four are ππ∗ states. Considering Table I, there aretwo interesting immediate observations: First, the number ofpromoted electrons p, as defined by the spatial integral overthe attachment or detachment density,30 is around 1.4 for thenπ∗ states while it is close to one for all the ππ∗ states. Thisremains true even though all cases show predominant singleexcitation character (the squared norm T2 of the doubles am-plitudes is below 0.1 and the squared Frobenius norm ! ofthe 1TDM is always above 0.85) with little variation. Thiscounterintuitive phenomenon, i.e., how more than one elec-tron can be “promoted,” without apparent double excitationcharacter, will be further discussed below. The second obser-vation is that each nπ∗ state can be described by a single or-

Hole Particle

(a) (b)

ω = 0.852 ω = 0.852

(c) (d)

λ1 = 0.849 λ1 = 0.849

(e)

FIG. 3. Analysis of the transition density matrix of the 11A2 state of pyridine:(a) hole ρH and (b) particle ρE densities (isovalue 0.005 e), (c) hole ψ0I

1 and(d) particle ψI0

1 NTOs, and (e) transition density ρ0I.

bital excitation (PRNT O ≈ 1) while at least two independenttransitions are required to describe the individual ππ∗ states(PRNT O ≈ 2).

In the following, the 11A2 state is analyzed in more de-tail as an exemplary nπ∗ state. This state is described by onlyone significant transition when considering the canonical MObasis (11a1 → 2a2 with a weight of 88%, see Figure S1 ofthe supplementary material29). The same transition also con-stitutes the only significant NTO pair (with an amplitude of λ1= 0.85 ≈ !). The NTOs ψ0I

1 and ψ I01 , the resulting hole (ρH)

and particle (ρE) densities as well as the transition density(ρ0I) are presented in Figure 3. The shapes of these objectsclearly reflect the relations given in Paper I1 (Eqs. (56)– (58))

ρH (r) = λ1ψ0I1 (r)2, (1)

ρE(r) = λ1ψI01 (r)2, (2)

TABLE I. Excitation energies (&E, eV, oscillator strengths in parentheses), dipole moments (µ, D), type assignments, state-averaged NTO transitions, squarednorm of doubles amplitudes (T2), squared Frobenius norm of the 1TDM (!), NTO participation ratio (PRNT O ), and number of promoted electrons (p) of thefirst six singlet excited states of pyridine computed at the ADC(2)/cc-pVDZ level of theory.

State &E µ Type Transition T2 ! PRNT O p

11B2 5.15(0.00) 3.35 nπ∗ ψa1,H → ψb2,E 0.09 0.86 1.01 1.41

11A2 5.40(0.00) 2.90 nπ∗ ψa1,H → ψa2,E 0.10 0.85 1.01 1.45

11B1 5.47(0.02) 0.46 ππ∗ ψa2,H → ψb2,E / ψb2,H → ψa2,E 0.10 0.85 1.77 1.07

21A1 6.95(0.03) 1.16 ππ∗ ψa2,H → ψa2,E / ψb2,H → ψb2,E 0.06 0.88 1.88 1.01

21B1 7.79(0.60) 1.37 ππ∗ ψb2,H → ψa2,E / ψa2,H → ψb2,E 0.08 0.89 1.95 1.12

31A1 7.89(0.55) 0.33 ππ∗ ψb2,H → ψb2,E / ψa2,H → ψa2,E 0.08 0.89 2.33 1.10

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024107-4 Plasser et al. J. Chem. Phys. 141, 024107 (2014)

Detachment Attachment

pD = −1.450 pA = 1.450

d1 = −0.967 a1 = 0.942

d2 = −0.157 a2 = 0.165

d3 = −0.095 a3 = 0.092

FIG. 4. Attachment/detachment densities (computed from the difference1DM, isovalue 0.005 e) and the major constituting NDOs for the 11A2 stateof pyridine.

ρ0I (r) =!

λ1ψ0I1 (r)ψ I0

1 (r), (3)

under the assumption that all NTO amplitudes except for λ1vanish.

The analysis of the difference density matrix of the 11A2state (Figure 4) is more involved. The primary NDOs corre-spond to the main transition seen in the NTO or canonicalrepresentation, and the eigenvalues (d1 = −0.97, a1 = 0.94)indicate that indeed one electron was transferred in this way.However, there are additional significant contributions lead-ing to an overall value of p = −

"idi =

"iai = 1.45. A de-

tailed examination of Figure 4 suggests that these are relatedto orbital relaxation effects. While the primary transition takesaway an electron from the nitrogen, there are secondary pro-cesses restoring electrons around this atom. These processesmay be quantified using a Mulliken analysis, which revealsthat the N-atom contributes 0.655 electrons to the detachmentdensity, but also receives 0.331 electrons in the formal attach-ment process. This relaxation effect is probably related to anextended charge shift in this state (as seen by the dipole mo-ment µ = 2.90 D), see, e.g., Ref. 31. Finally, it is interesting tocompare the pictorial representation of the attachment density(Figure 4) with the one of the particle density (Figure 3): the

particle density closely resembles the relevant π∗ MO whilethe attachment density possesses a notable additional contri-bution on the N-atom, in accordance to the previous discus-sion.

When analyzing excited state character it is commonpractice to regard the response vectors of correlated excitedstate methods as if they referred to individual Slater deter-minants. Such an approach is certainly sufficient for manyqualitative purposes (and it has even been used to derive ap-proximate non-adiabatic interactions32). However, the presentexample highlights that the underlying wavefunction struc-ture is indeed much more complex (the ADC response vec-tor refers to correlated intermediate states33) and may con-tain relaxation contributions, which are not directly seen inthe response vector. NDOs provide a convenient tool to un-cover these phenomena. Furthermore, this type of analysiscarries potential to provide deeper insight into the theory ofenergy gradients,34 where orbital relaxation modifies the at-tachment/detachment densities,30 and into the performance oforbital-relaxed excited state methods.35

As a second example, the 11B1 (ππ∗) state is briefly dis-cussed. In this case the different types of analyses producesimilar outcomes. The NDOs (Figure S4 of the supplemen-tary material29) and NTOs (not shown) are very similar tothe canonical MOs and the in-depth analysis performed abovewould not yield any additional information. However, as thisstate cannot be described by a single NTO pair (PRNT O ≫ 1),a different level of complexity is encountered, which is relatedto static correlation36 or entropy.37, 38 Additional insight intothese phenomena may be obtained through the analysis of thecharge transfer numbers !AB or in terms of localized molec-ular orbitals. However, a more detailed discussion of theseideas is postponed to the model of six neon atoms (Sec. IV)in which the relevant phenomena are more clearly visible.

The excited states of pyridine were recomputed at theADC(3) and MR-CISD levels of theory to evaluate themethod dependence of the reported results (Table II). Com-parison to ADC(2) shows that the 11A2(nπ∗) state is some-what increased in energy at these higher levels of theory.However, aside from this energetic difference, there is a re-markable agreement of the !, PRNT O , and p values across allthree computational methods suggesting that these are indeedvaluable universal descriptors, independent of the wavefunc-tion model. The NDOs and attachment/detachment densities

TABLE II. Excitation energies (&E, eV, oscillator strengths in parentheses)and different density matrix based descriptors (see text) of the first six singletexcited states of pyridine computed at the MR-CISD and ADC(3) levels oftheory (state ordering according to ADC(2) energies).

ADC(3)/cc-pVDZ MR-CISD(6/8)/cc-pVDZ

State &E ! PRNT O p &E+P ! PRNT O p

11B2 5.12(0.01) 0.82 1.02 1.29 5.15(0.01) 0.85 1.00 1.3411A2 5.81(0.00) 0.83 1.01 1.38 5.52(0.00) 0.85 1.00 1.4711B1 5.19(0.02) 0.80 1.75 1.09 5.17(0.03) 0.88 1.84 1.0021A1 6.72(0.01) 0.86 1.99 1.06 7.02(0.01) 0.87 1.96 1.0221B1 7.68(0.53) 0.84 1.96 1.16 7.85(0.75) 0.90 1.97 1.1231A1 7.80(0.57) 0.86 2.26 1.11 7.91(0.81) 0.93 2.14 1.09

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024107-5 Plasser et al. J. Chem. Phys. 141, 024107 (2014)

of the 11A2(nπ∗) state at the MR-CISD level are presented inFigure S5 of the supplementary material.29 These possess ingeneral a very similar shape as compared to the ADC(2) re-sults with the only exception that at the MR-CISD level thereis a small additional detachment contribution at the C atom inpara position to the N atom. These small differences can pro-vide cues about differences and possible shortcomings of thewavefunction models applied. A more detailed investigationof these effects would certainly be of interest but is out of thescope of this work.

IV. CHARGE RESONANCE INTERACTIONS IN A Ne6MODEL

Electron-hole correlations and the resulting charge reso-nance interactions are crucial properties of excited states inorganic semi-conductors39 and play an important role in ex-cimer formation.40, 41 However, understanding such effects inthe context of typical quantum chemical calculations is a ma-jor conceptual challenge. Therefore, an artificial model sys-tem is constructed here which allows to highlight the rele-vant effects in a systematic manner. This system consists ofsix weakly interacting neon atoms (following Ref. 42 wherea similar system was studied in a time-dependent fashion)which are arranged as a regular hexagon using a variable in-teratomic separation d (Figure 1(b)). At larger d values theatoms act as isolated chromophores and the excited states canbe described as coupled local excitations. Translated into theelectron-hole picture this situation is termed “Frenkel exci-ton”: a state where electron and hole move in a concertedfashion in order to be located at the same fragment at any onetime. At smaller separations, orbital interactions come intoplay and charge transfer configurations mix with the local ex-citations. This means that electron and hole move more freelyleading to increased charge separation.

Before discussing the excited states in more detail, itshould be mentioned that the system possesses D6h symme-try. Consequently, all orbitals and states are evenly delocal-ized over the whole system (except for degenerate ones). Itis therefore challenging to deduce any information about thewavefunction with standard methods. Conversely, the data ofinterest are coded in the relative motions of electron and holeas described by the two-body exciton wavefunction χ exc(rH,rE) (see Paper I1) and is therefore also not easily amenable fora pictorial representation. To overcome this problem, a math-ematical analysis in terms of the charge transfer numbers !ABis used, which are partial sums over 1TDM blocks (see Eq.(51) of Paper I1).40 First, the charge transfer ratio

ωCT = 1!

#

B =A

!AB (4)

is computed where A and B run over individual neon atoms.This measure is zero for local excitations and Frenkel excitonswhile it may reach a maximum value of one in the case ofcompletely charge separated states. In addition, a coherencelength

ωCOH ="

A

$"B !AB

%2 +"

B

$"A !AB

%2

2"

A,B !2A,B

(5)

is defined, using a formula, which is slightly modified with re-spect to, both, Refs. 40 and 43. ωCOH can vary between 1 andthe number of fragments N. The minimal value of ωCOH = 1is obtained for a diagonal " matrix (Frenkel exciton) or moregenerally under the condition that the " matrix possessesat most one non-vanishing value per row and column. Con-versely, the maximal value of ωCOH = N is reached in thecase of !AB = 1/N2 for all pairs of fragments A and B, thecase where electron and hole are uncorrelated and distributedover the whole molecule. In addition, the NTO participationratio (PRNT O) as given in Paper I1 (Eq. (59)) will be used as ameasure for the minimal number of configurations needed todescribe the excited state.

The lowest excited state of an individual neon atom isdescribed as a 2p → 3s Rydberg transition. Considering thatthere are three such transition per atom, the lowest excitedlevel of Ne6 at infinite interatomic separation is 18-fold de-generate. At lower separation it is expected that these 18 statesinteract among each other and mix with an even larger num-ber of charge transfer states. A full discussion of this manifoldis out of the scope of this work and only the S1 state will beused to exemplify the processes occurring. To examine thenature of this state, a separation of d = 3.0 Å is chosen as afirst example, and the hole and particle densities, NTOs, andtransition density of this system are presented in Figure 5.Several NTOs deriving from different symmetry adapted lin-ear combinations (SALC) of the atomic orbitals are involved.The presence of rather strong orbital interactions in this casecan be seen in the fact that out of the six possible SALCs,only three contribute significantly while the other three onlyshow a secondary involvement leading together to a valueof PRNT O = 4.14. The plot of the matrix of charge transfernumbers " is shown in Figure 5(f). The dark grey boxes onthe diagonal (going from lower left to upper right) show thedominance of local excitation character. However, the lightgrey boxes show that there are also important charge reso-nance interactions between neighboring atoms (note that thelight grey boxes in the upper left and lower right corners arepresent because of the circular structure of this system). Theseamount to 24% of the total excitation (ωCT = 0.24) and in-duce a non-negligible coherence length of ωCOH = 1.73. Bycontrast, at a separation of d = 5.0 Å the orbital interactionstrength is almost vanishing (Figure S6 of the supplementarymaterial29). Six NTOs contribute with almost equal weightsof λi ≈ 1/6 (PRNT O = 5.96), and there is almost no inter-atomic coherence (ωCOH = 1.02) or charge transfer character(ωCT = 0.01).

The dependence of the energy and the 1TDM descrip-tors of the S1 state (computed at the ADC(2) level) on theintermolecular separation d is presented in Figure 6. Thesystem exhibits an exciplex type energy profile, i.e., it isstrongly bound in the excited state (by about 0.8 eV at a dis-tance of d ≈ 3.0 Å) while there is almost no binding interac-tion in the ground state (<0.05 eV). The 1TDM descriptorsfollow the trends of the two examples presented above. Atd = 6 Å the different descriptors possess the idealized valuesfor six coupled local transitions: ωCT = 0.00, ωCOH = 1.00,and PRNT O = 6.00. When the separation is decreased, or-bital interactions come into play, which lead to an increase in

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024107-6 Plasser et al. J. Chem. Phys. 141, 024107 (2014)

FIG. 5. Analysis of the ADC(2) transition density matrix of the first singletexcited state of the hexagonal Ne6 system at d = 3.0 Å: (a) hole and (b) par-ticle densities (isovalue 0.0004 e), (c) hole and (d) particle NTOs (isovalue0.02 e), (e) transition density, and (f) plot of the " matrix.

ωω

FIG. 6. Properties of the first singlet excited states of the Ne6 model systemat different intermolecular separations d: (a) electronic energy relative to theground state at infinite separation and (b) different 1TDM based descriptors.

ωCT and ωCOH. This change in wavefunction character is alsoreflected through a decreasing PRNT O value. The last pointconsidered (d = 1.8 Å) exhibits the values of PRNT O = 2.21,ωCOH = 4.32, and ωCT = 0.64. This geometry is, however, al-ready in the strongly repulsive region and an idealized fullycoherent state cannot be observed in this geometric range.

In Figure 6, an inverse relationship between ωCOH andPRNT O is found. Such a connection is at first sight unexpectedas it occurs between two quantities, which are constructed inentirely different ways (one through a population analysis andthe other one from the singular value spectrum of the 1TDM).However, it is indeed possible to find a connection betweenωCOH and PRNT O through the block structure of the 1TDM:As mentioned above, a value of ωCOH = 1 for a delocalizedstate can only be obtained through a sparse structure of the" matrix, which necessarily derives from a blocked struc-ture of the 1TDM. If, in turn, the 1TDM possesses N sepa-rate blocks, there has to be at least one non-vanishing singu-lar value per block, which means that PRNT O ≥ N . A moredetailed derivation of this relation in the case of N = 2 can befound in the supplementary material of Ref. 40.

The above discussion highlights the importance ofPRNT O as a gauge for delocalization between chromophores(see also Ref. 44). Stated more simply, the value of PRNT O

= N shows that N independent local transitions are needed toform a delocalized Frenkel exciton. An additional interestingaspect is that the number of non-vanishing NTO amplitudeshas been related to static correlation.36 Also this idea is inagreement with the electron-hole picture: at large separations(right side of Figure 6 where PRNT O ≈ 6) electron and holemove in a correlated fashion in the sense that they are alwayson the same fragment at any one time. When lowering the in-teratomic distance (left side of Figure 6) charge resonance in-teractions come into play, which lift this restriction and allowelectron and hole to move more freely, i.e., less correlated,and accordingly PRNT O is lowered. These concepts followour previous conclusions, drawn in the cases of excimers be-tween naphthalene molecules40 and DNA bases41, 45 and theimportance of charge transfer interactions for excimers hasindeed been shown by experiment.46, 47

V. DOUBLE EXCITATIONS IN CONJUGATEDPOLYENES

To test the generality of the presented approach, a conju-gated polyene (all-trans hexatriene) was chosen as an examplewith excited states possessing significant double excitationcharacter. Before starting the discussion, it should be men-tioned that aside from normal point group symmetry, polyeneexcited states are classified as either ionic (+) or covalent (−).And both classes are notoriously difficult to describe for dif-ferent reasons. The former requires extensive treatment of σπ

correlations, while some of the latter possess extended doubleexcitation character. The different aspects of these featureshave been discussed in detail elsewhere.22, 48–51 In this work,the newly implemented ADC(3) method21 was used to allowan extensive treatment of correlation effects and a reliable de-scription of double excitation character. The results are com-pared to MR-CISD.

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024107-7 Plasser et al. J. Chem. Phys. 141, 024107 (2014)

TABLE III. Excitation energies (&E, eV), squared norm of doubles ampli-tudes (T2), and different density matrix based descriptors (see text) of theground and first three singlet excited states of all-trans hexatriene computedat the ADC(3) and MR-CISD levels of theory.

ADC(3)/cc-pVDZ MR-CISD(6/6)/cc-pVDZ

State &E T2 nu nu,nl ! p &E+P nu nu,nl ! p

11A−g 0 – 1.10 0.14 – – 0 1.08 0.24 – –

13B−u 2.40 0.10 2.99 2.17 0.88 1.16 2.75 2.81 2.18 – 1.22

13A−g 4.06 0.07 3.05 2.54 0.90 1.07 4.41 2.88 2.60 – 1.18

21A−g 4.55 0.77 3.13 2.41 0.22 1.83 5.52 3.37 2.74 0.38 1.60

23B−u 5.24 0.06 3.02 2.30 0.91 1.04 5.54 2.90 2.45 – 1.21

11B+u 5.61 0.08 3.02 2.13 0.88 1.09 5.84 2.74 2.08 0.87 1.00

11B−u 6.03 0.65 3.95 3.20 0.33 1.73 6.73 3.60 3.15 0.43 1.47

In Table III, specific information about the lowest foursinglet and three triplet states is presented. At the ADC(3)level, the T2 values (i.e., the squared norm of the doublesamplitudes) clearly show the double excitation character ofthe 21A−

g and 11B−u states. However, at the MR-CI level a

comparable measure cannot be readily defined, consideringthe different many-electron basis (intermediate states vs. con-figuration state functions) and the multireference formalism.Therefore, the squared Frobenius norm ! of the 1TDM isexamined as an alternative measure of the double excitationcharacter, which is well defined independent of the wavefunc-tion model. At the ADC(3) level T2 and ! correlate well witheach other in the sense that 0.95 ≤ T2 + ! ≤ 1 holds for allstates. Comparing ADC(3) with MR-CISD,52 there is an over-all good agreement of ! between these methods with the ex-ception that the double excitation character is somewhat morepronounced at the ADC(3) level.

In light of Sec. III, it is also interesting to analyze thepromotion numbers p and the NDOs. However, in this caseno significant secondary relaxation effects can be discoveredand the promotion numbers are already determined by thedouble excitation character in the sense that p ≈ 2 − !

≈ 1 + T2. Therefore, no further analysis was performed in thiscontext.

In Paper I,1 two measures for the number of unpairedelectrons53, 54 have been discussed: The linear nu value count-ing dynamic and static correlation effects, and the nu,nl value,which is intended to reduce the “noise” from dynamic corre-lation and focus on truly open-shell electrons. Using nu, theground state already possesses one effectively unpaired elec-tron, while nu,nl is significantly lower giving 0.14 for MP2(i.e., the ADC(3) ground state wavefunction in the currentimplementation) and 0.24 for MR-CISD. All singly excitedstates show an nu value rather close to three and a somewhatlower nu,nl value. This holds for MR-CISD as well, but withgenerally somewhat lower nu values. The largest nu value isfound for the doubly excited 11B−

u state (3.95 for ADC(3)and 3.60 for MR-CISD). By contrast, the other doubly excitedstate (21A−

g ) exhibits nu and nu,nl values only somewhat largerthan the singly excited states. This is probably related to thefact that its leading configuration is of HOMO2 →LUMO2

type, which has closed-shell character.

α β

n23,α = 0.022 n23,β = 0.014

n22,α = 0.974 n22,β = 0.019

n21,α = 0.978 n21,β = 0.972

n20,α = 0.978 n20,β = 0.973

n19,α = 0.979 n19,β = 0.978

FIG. 7. Comparison of the α and β NOs of the hexatriene radical cation inthe ground state.

VI. SPIN-POLARIZATION IN CONJUGATED POLYENERADICAL CATIONS

The analysis methods described in Paper I1 may be read-ily applied to spin-unrestricted calculations as well. In suchcases it is possible to construct either separate quantities forα and β densities or to analyze the spin-averaged and spin-difference densities. As an example, the hexatriene radicalcation is studied at the UADC(2) level of theory (for moreinformation on polyene radicals, see Ref. 24).

In Figure 7, the individual natural orbitals, i.e., the eigen-vectors of the α and β density matrices, computed for theMP2 ground state are presented. As opposed to the unre-stricted Hartree-Fock (UHF) reference orbitals there is nocomplete distinction between unoccupied and occupied or-bitals and the NOs possess fractional occupation numbers. Itis, however, clearly possible to identify strongly and weaklyoccupied orbitals, which derive from the occupied and unoc-cupied UHF orbitals, respectively. The first apparent differ-ence between the α and β NOs is of course the number ofstrongly occupied orbitals (22 α and 21 β). But there are alsosome notable differences in the shapes and occupation num-bers of the orbitals deriving from spin-polarization effects.The highest strongly occupied α NO contributes electron den-sity in particular to the central two carbon atoms (C3 and C4),while the corresponding unoccupied β NO shows a more evendistribution. On the α side the next lower strongly occupiedNO possesses σ character and an occupation number of n21,α= 0.978. By contrast, on the β side two π orbitals with some-what lower occupation numbers appear before the analogousσ orbital.

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024107-8 Plasser et al. J. Chem. Phys. 141, 024107 (2014)

TABLE IV. Excitation energies (&E, eV) at the UADC(2) level (oscillatorstrengths in parentheses), and different density matrix based descriptors (seetext) of the ground and first two singlet excited states of the all-trans hexa-triene radical cation.

State &E nu nu,nl !α !β pα pβ

1Au 0 2.10 1.19 – – – –1Bg 2.79(0.03) 2.56 1.95 0.26 0.63 0.44 0.802Bg 3.71(0.88) 2.63 2.01 0.64 0.28 0.77 0.39

In Table IV information about the three lowest states ofthe molecule is presented. The nu,nl value of the ground stateis close to 1 corresponding to the expected idealized behaviorand for both excited states nu,nl ≈ 2. Just as in the case of neu-tral hexatriene and other previously considered molecules18, 19

nu is significantly higher than nu,nl as different contributionsfrom dynamic correlation effects are included. With respect tothe excitation process, it can be clearly seen that for the 1Bgstate the predominant contribution is in the space of β orbitals(!β > !α and pβ > pα) while the opposite is true for the 2Bgstate. After spin-averaging one obtains in both cases ! ≈ 0.9and p ≈ 1.2. This corresponds to the situation of singly ex-cited states with some partial secondary orbital relaxation.

The spin densities (the difference between the α and β

densities, Fig. 8) allow for the visualization of the radicalcharacter of the different electronic states. For the groundstate (1Au) the major positive component is located on theouter carbon atoms (C1 and C6; 0.67 e per atom accordingto a Mulliken analysis), while C2 and C5 are predominantlypopulated with β spin (−0.32 e). The inner two C atoms (C3and C4) again contain excess α spin (0.23 e). A somewhatdifferent picture is present for the 1Bg state where the spin-density is more strongly localized at the outer two C-atoms(0.45 e) with only smaller contributions for the other atoms.The 2Bg state is similar to the ground state, only that an addi-

1Au

1Bg

2Bg

FIG. 8. Spin-difference densities (isovalue 0.005 e) of the first three loweststates of the hexatriene radical cation.

tional nodal plane is present, which is related to the promotionfrom NO(22, α) to NO(23, α) (cf. Figure 7).

VII. CHARGE TRANSFER STATES IN TRANSITIONMETAL COMPLEXES

Organic iridium complexes have attracted a great deal ofattention as efficient phosphorescent light emitters.55–57 How-ever, the description of the excited states of these systemswith wavefunction based ab initio methods proved to be quitechallenging.58 The purpose of this section is to shed morelight on this problem by means of a detailed analysis of theexcited state wavefunctions. The importance of relaxation ef-fects (p ≫ 1) is studied as these may explain why extendedwavefunction models are necessary to describe these systemscorrectly. A second point examined is the quantification ofmetal-to-ligand charge transfer (MLCT) character, which isof interest in the sense that, on the one hand, partial MLCT isnecessary for spin-orbit coupling enabling phosphorescence59

while, on the other hand, ligand centered character increasesthe magnitude of the transition moment. The two appliedquantification methods for MLCT (using the 1TDM and dif-ference density matrix) produced somewhat discrepant resultshighlighting an additional complexity in the quantitative com-putational analysis of such complexes.

The model complex fac-tris(3-iminoprop-1-en-1-ido)iridum abbreviated Ir(C3H4N)3, as shown in Figure 1(d),was chosen for this examination, which is a reducedversion of the well-known highly phosphorescent fac-trisphenylpyridinato iridium complex (Ir(ppy)3).60 Thissystem possesses C3 symmetry, therefore orbitals and statescan be classified according to A and E irreducible representa-tions, where the latter are doubly degenerate. The first nineexcited states of this complex (three of A and three pairs of Esymmetry) were computed. Selected data of these states aresummarized in Table V while the SA-NTOs are presentedin Figure 9. In the SA-NTO representation, three hole andthree electron orbitals suffice to describe all nine states. Theircharacter can be readily identified as Ir-d and ligand-π∗,respectively, which directly indicates that the inspected statespossess predominant MLCT character. By contrast, the HForbitals (Figure S7 of the supplementary material29) do notprovide such a clear picture. The occupied frontier orbitalspossess mixed Ir-d and ligand-π character while the virtualfrontier orbitals are mixtures between ligand-π∗ and diffuseorbitals. In terms of these orbitals all the excited states aredescribed by a number of different transitions. Thus, usingonly the HF orbitals, it is practically impossible to determinethe precise character of these states.

For a quantitative measure of MLCT character, the " ma-trix (as computed from the 1TDM) is considered and its de-tachment

ωD(Ir) = 1!

#

B

!Ir,B (6)

and attachment

ωA(Ir) = 1!

#

B

!B,Ir (7)

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024107-9 Plasser et al. J. Chem. Phys. 141, 024107 (2014)

TABLE V. Excitation energies (&E, eV) at the ADC(2) level, Mulliken net charges on the Ir atom (nI r ), SA-NTO transitions (with X2ij > 0.1), and different

density matrix based descriptors (see text) of the first nine singlet excited states of Ir(C3H4N)3.

State &E nI r ! ωD/A(Ir) PRNT O SA-NTO transitionsa p pD/A(Ir)

11A 0 0.12 – – – – – –21A 2.48 0.50 0.84 0.79/0.07 1.03 ψ1,H → ψ3,E 1.71 0.91/0.5211E 2.82 0.48 0.85 0.76/0.14 1.44 ψ1,H → ψ1,E/ψ1,H → ψ2,E 1.65 0.85/0.4821E 2.95 0.49 0.84 0.76/0.08 1.46 ψ2,H → ψ3,E/ψ1,H → ψ2,E 1.64 0.84/0.4631A 3.24 0.46 0.84 0.74/0.13 2.05 ψ1,H → ψ1,E/ψ3,H → ψ1,E/ 1.54 0.80/0.46

ψ2,H → ψ1,E/ψ3,H → ψ2,E

31E 3.35 0.46 0.85 0.73/0.16 2.05 ψ3,H → ψ2,E/ψ2,H → ψ1,E 1.57 0.82/0.4741A 3.43 0.43 0.85 0.73/0.17 2.10 ψ2,H → ψ1,E/ψ3,H → ψ2,E/ 1.56 0.81/0.49

ψ2,H → ψ2,E/ψ3,H → ψ1,E

aOnly one state for each set of degenerate states of E symmetry considered.

components are computed with respect to the iridium atom.For all states shown ωD(Ir) > 0.7 and ωA(Ir) < 0.2, whichhighlights the MLCT character. Interestingly, the analogouspA(Ir) and pD(Ir) values, which derive from a Mullikenanalysis of the attachment/detachment densities (see alsoSec. VI B of Paper I1), differ significantly from these val-ues. Specifically, the pA(Ir) values are enhanced notablywhile the pD(Ir) values are only somewhat larger than theirωD(Ir) counterparts. This results in a lowering of the netcharge transfer character computed from the difference den-sity matrix as compared to the 1TDM. The discrepancy be-tween these values raises the question, which type of anal-ysis produces the “better” result. In a sense, it can be ar-gued that there is simply no unique answer describing thecorrelated many-particle system. However, for practical us-age it is advisable to adjust the analysis to the main prop-erty of interest. If, on the one hand, transition properties(e.g., oscillator strengths) are studied, then it is more suit-

Hole Particle

ψ1,H(2.50) ψ1,E(2.51)

ψ2,H(2.49) ψ2,E(2.51)

ψ3,H(2.49) ψ3,E(2.48)

FIG. 9. State-averaged natural transition orbitals for the first nine singlet ex-cited states of the complex Ir(C3H4N)3. Summed amplitudes (λi,H , λi,E ) aregiven as well.

able to analyze the 1TDM. If, on the other hand, stateproperties (e.g., dipole moments, atomic populations) are ofinterest, then the attachment/detachment representation de-riving from the state densities may yield the more validoutcome.

The PRNT O values are presented in Table V as well.These reflect the degeneracy of the underlying orbitals: Forthe 21A state deriving from a simple a → a transition it holdsthat PRNT O ≈ 1 while for the 11E and 21E states, formed bya → e and e → a transitions (i.e., either the hole or parti-cle level is degenerate), PRNT O is about 1.5. Finally, in thecase of the 31A, 31E, and 41A states, which form an E × Erepresentation (reached by e → e transitions) PRNT O ≈ 2.

The number of promoted electrons as given by p is quitelarge (>1.5) in all cases. However, these states do not pos-sess any significant double excitation character (! ≈ 0.85,and also the T2 norms are all around 0.12). In a similar senseto Sec. III this can be identified with secondary orbital re-laxation effects. To understand these electron shifts in moredetail, the attachment/detachment densities of the 21A state aswell as the constituting NDOs are plotted in Figure 10. Thedetachment density shows more or less the expected shapeof a dz2 orbital on iridium with only some smaller contribu-tions around it. Interestingly, aside from the expected ligand-π∗ orbitals, the attachment density possesses a strong compo-nent on the Ir atom, as well. This situation is reflected by theNDOs. The first pair of NDOs (with eigenvalues d1 = −0.965and a1 = 0.923) relates simply to the Ir-d and ligand-π∗ or-bitals that are also seen in the SA-NTO analysis. However,there are four more significant contributions. On the attach-ment side these have the appearance of Ir-d orbitals, whiletheir detachment counterparts exhibit some Ir-ligand bondingcharacter. The occurrence of attachment Ir-d orbitals, whichare already occupied in the ground state, may be somewhatcounterintuitive. However, it should be remembered that thisdoes not relate to the primary excitation but to relaxation pro-cesses. A likely interpretation of this effect is a contraction ofthe Ir-d orbitals following the removal of an electron. Thisphenomenon is probably related to the double-shell effect,which has been discussed in the context of multi-referencecalculations (i.e., two sets of d-orbitals have to be included inthe active space).61

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024107-10 Plasser et al. J. Chem. Phys. 141, 024107 (2014)

Detachment Attachment

pD = −1.711 pA = 1.711

d1 = −0.965 a1 = 0.923

d2 = −0.108 a2 = 0.110

d3 = −0.108 a3 = 0.110

d4 = −0.083 a4 = 0.086

d5 = −0.083 a5 = 0.086

FIG. 10. Attachment/detachment densities and the major constituting NDOsfor the S1 state of Ir(C3H4N)3.

VIII. CONCLUSIONS

In the first part of this series1 a number of tools for theanalysis of electronic excitations, including several new meth-ods, were discussed and their implementation reported. In thiswork, five distinct examples were chosen: pyridine as a proto-typical heteroaromatic molecule, six interacting neon atomsfor studying charge resonance interactions, hexatriene in itsneutral and radical cation form as examples for two-electronexcitations and spin-polarization, respectively, and the modeliridium complex Ir(C3H4N)3 as a representative metal or-ganic compound. It was highlighted that the presented tool-box of analysis methods is not only useful for a quick and

automatized analysis of data sets, but more importantly that itprovides deeper insight into excited state structure, revealinga number of phenomena, which are usually not considered inthe discussions. To study these systems, computational resultswere compared at the ADC(2), ADC(3), and MR-CISD lev-els of theory and a good agreement of the different descriptorsbetween these methods was observed.

The natural difference orbitals (the eigenvectors of thedifference density matrix) proved to be of particular inter-est. Usually only the weighted sum over these is consid-ered, yielding the attachment/detachment densities.30 How-ever, it was found that the individual NDOs give a more dif-ferentiated picture and that aside from representing the pri-mary excitation process, they allowed the visualization ofsecondary relaxation effects (in particular, for pyridine andIr(C3H4N)3). This effect was especially pronounced if thepromotion number p was significantly larger than one. Tothe best of our knowledge such an analysis has so far onlybeen performed for isolated atoms in the literature42 and itappears that the full potential of this approach to analyze com-plex excitations in molecular systems has not been exploitedyet.

Furthermore, the concept of electron-hole correlationleading to charge resonance effects was studied in the Ne6model system. In this case, it was shown that an inverse re-lation exists between the coherence length of the excitation(ωCOH) and the number of NTO pairs describing it (PRNT O).This non-intuitive connection was supported by mathematicalarguments. While this discussion was carried out on a some-what abstract level, new tools to visualize correlated electronand hole densities are currently being constructed. Moreover,the utility of the state-averaged natural transition orbitals, asintroduced in Paper I1 was highlighted using the examplesof pyridine and Ir(C3H4N)3. It could be shown that the SA-NTOs possess significant advantages over simple HF orbitalsin the cases of large orbital basis sets or other situations ofbadly resolved HF orbitals. Examples of several other numer-ical descriptors were shown as well: ! the squared norm ofthe 1TDM measuring the single-excitation character, nu andnu,nl as two measures for unpaired electrons. Pictorial repre-sentations of a number of different types of orbitals (natu-ral, natural transition, natural difference) and densities (spin-difference, transition, hole/particle, attachment/detachment)were presented and compared. A population analysis ofthese densities was used to collect information in a compactway.

Regarding this extended toolbox, the question arises un-der which circumstances such a detailed analysis is advisableand when a simpler consideration is sufficient. For this pur-pose we suggest to start with the PRNT O and p values. Sig-nificant electron-hole correlations can only occur in the caseof PRNT O ≫ 1 and only in this case will it be advantageousto perform the analysis presented in Sec. IV. Moreover, aspointed out above, the PRNT O value by itself already containsimportant information about delocalization and coherences.On the other hand, an analysis of the NDOs should only benecessary in the case of p ≫ 1. Such a situation may eitherarise from relaxation effects (present when ! ≈ 1) or doubleexcitations (! ≪ 1).

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024107-11 Plasser et al. J. Chem. Phys. 141, 024107 (2014)

While we have attempted to provide a diverse set of ex-amples, our discussion is certainly not exhaustive and manyother interesting phenomena may be uncovered by the pre-sented approach or similar strategies. An interesting ques-tion, which is postponed for future work, is how the presentedmethodology may be used to create new diagnostics for thereliability of quantum chemical methods. On the one hand, anexplicit consideration of the electron-hole wavefunction maybe used to construct a more rigorous charge transfer diagnos-tic for time-dependent density functional theory.62, 63 On theother hand, it will be interesting to study the effect of the pro-motion number on the reliability of different methods (i.e.,how well relaxation effects can be described). There are ofcourse a number of possible extensions to the presented meth-ods. While the current analysis is based on different types ofpopulation analysis, future work is aimed at the use of spa-tial multipole moments for an alternative description. Further-more, it would be interesting to compare the present static re-sults with similar studies in the time domain.42, 64 Finally, anextension to the two-particle transition density matrix65 maybe helpful for the analysis of doubly excited states and biex-citons.

ACKNOWLEDGMENTS

F.P. is a recipient of a fellowship for postdoctoral re-searchers by the Alexander von Humboldt foundation. S.A.B.is a fellow of the Heidelberg Graduate School of Mathemat-ical and Computational Methods for the Sciences. Computertime at the Compute Server of the Interdisciplinary Centerfor Scientific Computing, Heidelberg, is gratefully acknowl-edged.

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